Walks on uncountable ordinals and non-structure theorems for higher Aronszajn lines
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913541945032704 |
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| author | Inamdar, Tanmay Rinot, Assaf |
| author_facet | Inamdar, Tanmay Rinot, Assaf |
| contents | It is proved that if there is an $\aleph_2$-Aronszajn line, then there is one that does not contain an $\aleph_2$-Countryman line. This solves a problem of Moore and stands in a sharp contrast with his Basis Theorem for linear orders of size $\aleph_1$.
The proof combines walks on ordinals, club guessing, strong colourings of three different types, and a bit of finite combinatorics. This and further non-structure theorems for Aronszajn lines and trees are established for successors of regulars, successors of singulars, as well as inaccessibles. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_08757 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Walks on uncountable ordinals and non-structure theorems for higher Aronszajn lines Inamdar, Tanmay Rinot, Assaf Logic Primary 03E02, 03E75, 06A05. Secondary 03E05 It is proved that if there is an $\aleph_2$-Aronszajn line, then there is one that does not contain an $\aleph_2$-Countryman line. This solves a problem of Moore and stands in a sharp contrast with his Basis Theorem for linear orders of size $\aleph_1$. The proof combines walks on ordinals, club guessing, strong colourings of three different types, and a bit of finite combinatorics. This and further non-structure theorems for Aronszajn lines and trees are established for successors of regulars, successors of singulars, as well as inaccessibles. |
| title | Walks on uncountable ordinals and non-structure theorems for higher Aronszajn lines |
| topic | Logic Primary 03E02, 03E75, 06A05. Secondary 03E05 |
| url | https://arxiv.org/abs/2410.08757 |